Modified correlated measurement errors model for estimation of population mean utilizing auxiliary information

The existence of measurement errors cannot be avoided in practice. It is a prominent fact that the existence of measurement errors diminishes conventional properties of the estimators. A modified correlated measurement errors model has been proposed. Shalabh and Tsai (Commun Stat Simul Comput 46(7):5566–5593. 10.1080/03610918.2016.1165845, 2017) correlated measurement errors model is a particular member of the suggested modified model. In this article, we have tackled the estimation of population mean utilizing auxiliary information under modified correlated measurement errors model. We have developed ratio and product estimators and studied their properties in case of simple random sampling without replacement (SRSWOR) up to first order of approximation. It has been illustrated that suggested ratio and product estimators are more efficient than the conventional unbiased estimator as well as Shalabh and Tsai (Commun Stat Simul Comput 46(7):5566–5593. 10.1080/03610918.2016.1165845, 2017) ratio and product estimators under very realistic situations. An empirical study has also been performed to demonstrate the merits of the recommended estimators over other estimators.

example, usually the same survey personal collects data on study and auxiliary variables both and so it may not be reasonable to presume that the MEs in both the variables are independent.Rather, they will be dependent (i.e., correlated) and this dependence in MEs may arise due to the hidden intrinsic tendencies of the surveyor.For further illustration, readers are referred to Shalabh and Tsai 19 , pp. 5567-5568.Shalabh and Tsai 19 were the first who discussed the impact of correlated measurement errors (CME) over the performance of ordinary ratio and product estimators of population mean.Later Boniface et al. 20 , Bhushan et al. 21,22 and Kumar et al. 23 have evaluated the performance of some estimators of population mean under CME.
Taking motivation from Diana and Perri 24 and Shalabh and Tsai 19 work, we have developed a modified correlated measurement errors model.This paper is an effort towards developing ratio and product estimators under a modified correlated MEs model.
The remaining sections of this article are organized as follows: Shalabh and Tsai 19 correlated measurement errors Model's along with the ratio and product estimators have been introduced in section "Shalabh and Tsai (2017) correlated MEs model's characteristics".In section "Description of modified correlated MEs model and the proposed estimators", we have developed the Modified correlated MEs Model and the proposed the ratio and product estimators in this scenario.The properties of the suggested estimators are examined up to first order of approximation (foa).We have covered the bias and MSE comparisons of the suggested mean per unit, ratio and product estimators with the usual mean per unit as well as the ratio and product estimators given by Shalabh and Tsai 19 in sections "Bias comparisons of t R , t * R , t P and t * P " and "Comparison of MSEs of (t * R , t * P ) with (y, y * , t R , t P )", respectively.The theoretical efficiency conditions of proposed estimators were also obtained.In Section "Special Case", a special case of the recommended ratio and product estimators under modified correlated MEs was also discussed.
In Section "Empirical study ", an empirical study is also provided for assessing the efficiency of proposed estimators.In section "Simulation study ", a simulation study has also been performed in R software to strengthen the current study.The results and discussion followed by conclusion of the current study are summarized in sections "Results and discussions" and "Conclusion", respectively.

Shalabh and Tsai (2017) correlated MEs model's characteristics
Let � = (� 1 , � 2 , ..., � N ) be a finite population of size N and a sample of size n be selected from the population using SRSWOR scheme.Assume that the true value of the ith unit of is denoted by X i and Y i corresponding to the auxiliary and study variables, respectively.
But these true values are somehow not available and rather these are detected as y i and x i having MEs denoted by u i and v i , respectively.Shalabh and Tsai 19 assumed that these values can be expressed in additive form defined as: The MEs u i and v i are unobservable and assumed to have mean 0 (zero) and different variances σ 2 u and σ 2 v , respectively, with correlation coefficient ρ uv .Moreover it is reasonable assuming uncorrelated MEs to the true values.Suppose that µ Y and µ X are the population means, σ 2 Y and σ 2 X are the population variances, C Y and C X are the population coefficients of variation while ρ YX is the population correlation coefficient.Further, consider y = 1 x i as the sample means of the observed values.
Assuming known population mean µ X of the auxiliary variable X, Shalabh and Tsai 19 proposed ratio as well as product estimators for the population mean µ Y of the study variable Y given as: Assuming large enough population size N, the finite population correction (fpc) term is It is easy to see that y is an unbiased estimator of µ Y and its variance/mean squared error (MSE) is given as: The bias and MSE of t R and t P up to first order of approximation (foa), are respectively, given as: (1)

Description of modified correlated MEs model and the proposed estimators
We define the following correlated MEs model for expressing the observed values y * i and x * i in the additive form of true values (denoted by X i and Y i ) and the MEs (denoted by u i and v i , respectively) as: where (α, η) are constants to be determined the conditions over (α, η) so that the model ( 11) is superior to the Shalabh and Tsai 19 model defined in (1).
The sample means denoted by y * and x * under the model (11) are defined as: Estimators y * and x * can easily be proved as unbiased estimators of the population means µ Y and µ X , respectively.
The variances/MSEs of y * , x * and the covariance between y * and x * under SRSWOR ignoring fpc term, are respectively, given by From ( 4) and ( 12), we have MSE(y * ) < MSE(y) if ( 9) Similarly, from ( 5) and ( 13), we note that MSE(x R in the form of e * 0 and e * 1 , we have Assuming will be expandable.Expanding (22) up to power two of e ′ s , we have We obtain the bias of t * R up to foa by taking expectation of (23) which is given as: It is observed from (24) that the bias of t * R will vanish when sample size n is sufficiently large.Further, the bias of t * R at (24) can be re-expressed as: (15) i.e., if |α| < 1 Thus, under the conditions, β YX = R and αβ uv = ηR , the proposed ratio estimator t * R is almost unbiased.After squaring (23) and ignoring greater than power two terms of e * 's, we obtain The expectation of (26) provides the mean squared error of t * R up to foa given as: We note that if ρ uv is positive, then we select (α, η) in such a way that the quantity αη is also positive.Alter- natively, if ρ uv is negative, then we choose (α, η) in such a manner that the quantity αη is negative.We also note that (α, η) may also take the values of correlation coefficients ρ YX and ρ uv .Progressing similar to t * R , the following expressions for the bias and MSE of the product estimator t * P up to foa can be obtained as follows: Expression (29) clearly shows that the bias of t * P is zero, for sufficiently large n.The bias of t * P at (29) can be re-written as: The above expression clearly indicates that the product estimator t * P is unbiased if ρ YX = 0 and ρ uv = 0, i.e., if the correlation between the two variables Y and X is zero and the measurement error variables u and v are uncorrelated.
Further, the bias of the ratio estimator t * R (of the product estimator t * P ) decreases as the sample size n increases and can be easily seen that the proposed ratio (product) estimator t * R (t * P ) is consistent.We note from (32) that if ρ uv is positive, then to get large efficiency we select (α, η) in such a way that the quantity αη is negative.On the other hand, if ρ uv is negative, then we choose (α, η) in such a manner that the quantity αη is positive.
It can be noticed from the expressions of bias and MSE of the estimators (t R , t P , t * R , t * P ) that data having exist- ence of the MEs lead a supplementary term in each instance.However, this additional term disappears in case of no MEs on both the variables.
This study can also be extended on the lines of Shahzad et al. 25 and Ali et al. 26 .• From ( 7) and ( 24), we have that

Bias comparisons of
This inequality will meet usually in survey situations as |α| < 1 and |η| < 1.Now, we consider the two situations: Thus in this situation (ρ uv = 0), the suggested estimator t * R is less biased than the Shalabh 7 and Shalabh and Tsai 19 ratio estimator t R .Here we would like to mention that the properties of t R have been studied by Shalabh 7 in case of no correlation between the MEs.

Comparison of MSEs of (t *
R , t * P ) with (y, y * , t R , t P ) • From ( 4), ( 9), ( 12) and ( 27) it is noted that 1. the suggested estimator t * R is said to be more efficient than the conventional unbiased estimator y and the suggested estimator y * , respectively, if 2. The proposed estimator y * is more precise than Shalabh and Tsai 19  (a) if ρ uv = 0 , then inequalities (37)-( 40), respectively, reduce to: From (44) it is clear that when ρ uv = 0, the recommended ratio estimator t * R is always better than Shalabh 7 , and Shalabh and Tsai 19 ratio estimator t R , as in this case the inequality (44) always holds good.If K YX < 1 2 , then the inequality (43) holds true, i.e., the suggested estimator y * is said to be more efficient than Shalabh and Tsai 19 estimator t R , while for K YX < 1 2 , the inequality (42) does not hold good, i.e., the suggested ratio estimator t * R is inferior to the proposed estimator y * .If K YX < 1 2 , inequality (41) is not hard to meet in the survey situations which suggests that the offered ratio estimator t * R is better than the conventional unbiased estimator y.
(b) if η = α , then inequalities (37), ( 38) and ( 40), respectively, boils down to: • From ( 4), ( 10), ( 12) and (30), we observe that the offered product estimator t * P is better than the estimator: The proposed product estimator t * P will be more efficient than y and y * , if the conditions (56) and (57), respectively, hold good.Further the estimator y * is superior to the Shalabh and Tsai 19 product estimator t P as long as the inequality (59) satisfied.
provided that the ratio R = µ Y µ X is non-negative and (α, η) have the same signs.When there are no measurement errors in the auxiliary variable and/or the measurement errors associated with the study and auxiliary variables are not correlated, the condition (60) boils down to ρ YX > 0, which is usual condition derived under the specification of no measurement errors.
If we set α = η = 1 in (60), then we have which is due to Shalabh and Tsai 19 .
From ( 4) and (66), we have that Further from ( 12) and (66), we observe that Hence, the recommended estimator t * * R is better than y and y * , respectively, if the inequalities (68) and (69) are satisfied.

Empirical study
To judge the performance of the recommended estimators, we have performed an empirical study using two real populations earlier considered by Bhushan et al. 21,22 .
Population I: Source: Gujarati and Sangeetha (2007).Y i and y i are true and measured consumption expenditure, respectively.X i and x i are true and measured income, respectively.
Population II: Source: The book of U.S. Census Bureau (1986).Y i and y i are true and measured value of the product sold, respectively.X i and x i are true and measured size of farms, respectively.We have used the following formulae for computing percent relative efficiencies (PREs) of various estimators of µ Y with respect to y : These values are displayed in Tables 1, 2, 3 and 4. Computation time for the empirical study: We have noted down the computation time for the numerical study.The time taken for each value of α is 0.073 s while the total computation time is 1.022 s (= 0.073 × 14).
The biases and MSE of the estimators for no measurement errors case, i.e., σ 2 u = 0 and σ 2 v = 0 for n = 20 and 100 are noted in Table 5.The biases of all estimators are given in Tables 6 and 8 for n = 20 and 100, respectively while the MSEs of theses estimators are recorded in Tables 7 and 9 for n = 20 and 100, respectively, for various combinations of α, η, ρ XY and ρ uv .
(71) Computation time for the simuation study: We have noted down the computation time for the simulation study also.The time taken for one iteration (for each combination of α, ρ xy and ρ uv ) is 2.138 s while the total computation time is 4.632333 min (= 2.138 × (5 × 5 × 5 + 5))/60).

Results and discussions
From Tables 1, 2, 3 and 4, we observe the followings: 1.The proposed ratio estimator t * * R has bias very marginally larger than the ratio estimator t R in population-I while it (proposed ratio estimator t * * R ) is less biased than the ratio estimator t R for the population-II.Further, it is observed that the suggested product estimator t * * P is less biased (in the sense of absolute bias) than the product estimator t P for both the populations I and II. 2. The recommended unbiased estimator y * is more efficient than the conventional unbiased estimator y with marginal gain in efficiency for |α| < 1 in Populations I and II.

The recommended ratio estimator t * *
R is more efficient than y, y * , t P and Shalabh and Tsai 19 ratio estimator t R with considerable gain in efficiency under the condition |α| < 1 in Population I, while it is inferior to y and y * in Population II due to negative correlation between (Y &X) and R is superior to Shalabh and Tsai 19 Similarly, from Tables 5, 6, 7, 8 and 9, we can compare the biases and MSEs under both conditions without measurement error as well as in the presence of measurement error.From these Tables 5, 6, 7, 8 and 9, we note the followings: 1. Tables 5, 6, 7, 8, 9 clearly reveal the higher values of bias and variance or MSE under presence of measurement errors, i.e., σ 2 u = 1 and σ 2 v = 1 than the values under no measurement errors, i.e., σ 2 u = 0 and σ 2 v = 0 .Thus it indicates that the properties of estimators got affected by the presence of measurement errors.2. The proposed unbiased estimator y * is having less bias and MSE than the conventional unbiased estimator y for |α| < 1 and both sample sizes, i.e., n = 20, 100.3. From Tables 5 and 7, the bias of the suggested estimators t * * R and t * * P are compared in the presence of measurement errors.It can be clearly observed that bias of t * * R and t * * P are impacted by the value of ρ uv and these are substantially different for ρ uv = 0 and ρ uv = ± 0.9, indicating the significant impact of correlated measurement errors.Apparently, the bias decreases as sample size increases but there is no apparent reduction in the differences in the values of bias for ρ uv = 0 and ρ uv = ± 0.9.So, we can conclude that the correlated measurement errors influence the bias of the estimators compared to uncorrelated measurement errors.4. From Tables 6 and 8, we can observe a clear impact of sign of correlation between measurement errors on the MSE values of estimators t * * R and t * * P .The MSE of t * * R (in case of highly positively correlated study and auxiliary variable, i.e., ρ XY = 0.9) is lowest for positively correlated measurement error, i.e. for ρ uv = 0.9.The MSE of t * * R decreases As the degree of ρ XY increases for ρ XY > 0. However, the extent of ρ uv also affects the rate and value of MSE.Obviously the MSE decreases as sample size increases for all the values of the parameters considered for ρ XY > 0. 5.In the same way, we can conclude for the estimator t * * P (in case of highly negatively correlated study and auxiliary variable, i.e., ρ XY = -0.9) is lowest for negatively correlated measurement error, i.e. for ρ uv = -0.9.The MSE of t * * P decreases As the degree of ρ XY increases for ρ XY < 0. This clearly indicates that the presence of measurement errors affected the MSE of t * * R and t * * P .www.nature.com/scientificreports/6. Tables 5, 6, 7, 8 clearly depicts that the biases and MSEs of the suggested estimators are the lowest at α = η = 0.05.
Thus, the recommended ratio (t * * R ) and product (t * * P ) estimators are useful in practice.

Conclusion
This paper has introduced a modified correlated MEs model.The proposed correlated MEs model involves a constant α (say) with restriction |α| < 1, termed as 'error control parameter' .This error control parameter α (say) controls the errors in observations if we choose error control parameter α (say) near to 'zero' .For α = 1, proposed correlated MEs model reduces to Shalabh and Tsai 19 model.We have suggested ratio as well as product estimators for population mean ( µ Y ) of the study variable Y in presence of auxiliary variable X when correlated MEs contaminate the observations on both study and auxiliary variables.The expressions of bias and MSE of the recommended ratio and product estimators are determined up to foa under SRSWOR sampling scheme.The realistic conditions are derived under which the recommended ratio and product estimators act superior than the conventional unbiased estimators (y, y * ) and Shalabh and Tsai 19 ratio (t R ) and product (t P ) estimators.An empirical study and a simulation study have also been performed in R software to exhibit the performance of the recommended ratio and product estimators over usual unbiased estimators and the ratio and product estimators due to Shalabh and Tsai 19 .It is observed that when the 'error control parameter' is close to 'zero' , the recommended ratio and product estimators yield larger gain in efficiency.Thus, we recommend the proposed study for its use in practice.Table 6.Bias of the estimators for n = 20 and several values of α and η.
Here we note that α and η may take the values of ρ YX and ρ uv as |ρ YX | < 1 and |ρ uv | < 1.Now we define the ratio ( t * R ) and product ( t * P ) estimators for population mean µ Y of Y under the model(17)asTo study the properties of the estimators t * ) < MSE(x) if Thus, the resulting modified correlated MEs model is: with |α| < 1 and |η| < 1. * R and t * P under the model (17), we write We note that e * 0

t R , t * R , t P and t *
are unbiased whereas t R , t * R , t P and t * P are biased estimators of the population mean µ Y of Y.This fact holds correct whether the MEs exist or do not exist. * R is always less biased than the Shalabh and Tsai 19 ratio estimator t R .2. if ρ uv = 0 , i.e., MEs u i and v i are not correlated, then inequality (33) boils down to: which again holds good as |η| < 1.
Shalabh and Tsai 19 ) if It is further observed from (9) and (12) that MSE(y * ) < MSE(t P ) , if Now we discuss two cases: (c) if ρ uv = 0 , then conditions (48)-(51), respectively, reduce to: Inequality (54) clearly propagates that the recommended product estimator t * P is better than Shalabh and Tsai 19 product estimator t p as |α| < 1 and |η| < 1 .Further t * P is more efficient than y and y * * P is better than Shalabh and Tsai 19 product estimator t P as long as σ Thus the recommended product estimator t * * P is better than y and y * provided that the inequalities (70) and (71) satisfied, respectively.

Table 2 .
Variances/MSEs and PREs (with respect to y ) of y * for several values of α.
product estimator t P .4. The recommended product estimator t * * P is better than the estimators y, y * and Shalabh and Tsai 19 product estimator t P under the condition |α| < 1 in Population II, while it is inferior to the estimators y and y * in α Var (y * ) PRE (y * , y)

Table 3 .
Biases, MSEs and PREs (with respect to y ) of t * * R for several values of α.

Table 4 .
Biases, MSEs and PREs (with respect to y ) of t * * P for several values of α..

Table 7 .
MSE of the estimators for n = 20 and several values of α and η.

Table 9 .
MSE of the estimators for n = 100 and several values of α and η.